218 CHEMICAL ENGINEERING It will be noted that if eight of the half -diameter cubes be sheared to produce cubes of half their size, the new surface presented will be double that formed when the single large cube is sheared into half -size cubes, and also that the energy required in the case of the smaller cubes is double that required in the case of the larger cubes. This should demonstrate that energy applied to crushing is proportional to the surface produced. In this scheme we start from an infinite mass, say the side of a mountain, and consider that particles are sheared off in the form of cubes, each particle having new surface added to it equal to that exposed before the new shearing operation was started. This eliminates the "minus one" part of the equation given in Richards' "Ore Dressing" on the subject. For example, if we cut off from a cubical corner of this infinite mass 100 slabs 1 in. thick and 100 in. square, and then cut the slabs up into 1-in. cubes, 6,000,000 sq. in. of new surface is produced. If instead, 200 %-i&. slabs 170 Compression, Inches _0__0.04_ 0.08 .0.12' 0.16 0.20 0.24 018. 032 0.56 0.40 Crushed 3 times wnrt 'Inrer/neaf/afe screen-•mg removing-40 me material 120 160 200 240 280 520 360 400 440 Re'ciproca Is of Diameters (Inches) FIG. 15.—Diagram of crushing tests. 4&0 were cut from the same corner and likewise cut up into J^-in. cubes, the area would be. 12,000,000 sq. in., just double the surface of the 1-in. cubes. This shows that the surface produced on equal masses of rock is proportional to the reciprocal of the diameters, as can be proved in a similar manner for any sizes. Also the surfaces of two lots of the same sized particles will be proportional to their values, and if of the same substances, proportional to the weights of the lots. From the above we see that the energy absorbed by a lot of given sized particles is proportional to the product of their surface by their weight, which can be shown graphically by an area. And if we have a series of these areas, placed side by side, and representing the summation of the energies of the different sizes produced by a crushing operation, the total area is proportional to the energy expended on the rock. And when the cumulative weights of the different sizes are plotted consecutively, as in Fig. 14, the area between the sizing-analysis curve and the zero lines is proportional to the work expended on the rock in breaking it down from the infinite mass. And then if further crushing takes place on all or part of this rock and the new-sizing analysis is plotted, the area between the two curves is proportional to the further work done. This is the crushing-surface diagram. For rough work, comparing the work of one day with that of another, the screen analysis may be plotted, mesh against the percentages, and areas compared, etc. For accurate work, such as the determination of constants for different rocks, or for the comparison of the efficiencies of different crushing machines, etc., the actual